A Spectral Color Imaging System for Estimating Spectral Reflectance of Paint

Transcription

1 A Spectral Color Imagng System for Estmatng Spectral Reflectance of Pant Vladmr Bochko Department of Informaton echnology, Lappeenranta Unversty of echnology, P.O.Box 2, Lappeenranta, Fnland Normch sumura and Yoch Myake Department of Informaton and Image Scences, Chba Unversty, 1-33 Yayo-cho, Inage-ku, Chba , Japan ) In ths paper, the analyss methods used for developng magng systems estmatng the spectral reflectance are consdered. he system ncorporates the estmaton technque for the spectral reflectance. Several tradtonal and machne learnng estmaton technques are compared for ths purpose. he accuracy of spectral estmaton wth ths system and each estmaton technque s evaluated and the system s performance s presented. Introducton In ths paper, the analyss methods used for developng magng systems estmatng the spectral reflectance are consdered. he estmaton of the spectral reflectance determnes a performance of a hgh qualty color magng system whch s requred n dgtal archves, network museums, e-commerce and telemedcne. 1 Especally the desgn of a system for accurate dgtal archvng of fne art pantngs has awakened 1

2 ncreasng nterest. In such a system the dgtal mage s easly examned by usng a broadband network. he vstors of museums, art experts and artsts would be able apprecate a varety of pantngs at any vewng ste wherever those pantngs are located. In addton, archvng the current condton of a pantng wth hgh accuracy n dgtal form s mportant to preserve t for the future. Several research groups worldwde have been workng on these problems. 2,3,4,5,6,7,8,9,1,11,12,13,14 Conventonal color magng systems have a lmtaton that s a dependence of mages on the llumnant and characterstcs of the magng system. he magng systems based on spectral reflectance, unlke the conventonal systems, are devcendependent and capable of reproducng the mage of the scene n any llumnaton condtons. Also, these systems can ncorporate the color appearance characterstcs of the human vsual system. Owng to the fact that spectral characterstcs are smoothed, the hgh-dmensonal spectral reflectance s accurately represented by a small number of channel mages. 15,16,17 herefore, the task of spectral estmaton ncludes statstcal analyss of the reflectance spectra and mnmzaton of the estmaton error. he choce of error measures s a general topc of broader nterest and sometmes contrary n mpact. In the archval, ramfcatons for optmzng more for RMSE versus color dfference depend on applcatons. For example, spectral optmzaton may better enable the dentfcaton of colorants used whle color dfference optmzaton may yeld superor vsual reproductons. he tradtonal technques used for the estmaton nvolve matrx-vector computaton and usually assume a lnear model of the data. Although the approach based on lnear algebra and a nonlnear data model s proposed n the lterature, 4 machne learnng 2

3 technques seem appealng. hey estmate spectra of the scene, ncorporate the data nonlnearty and nvolve the tranng and predcton procedures. herefore, the neural networks based methods for spectral reconstructon are proposed by Rbes et al. 18 he tested methods are superor to the pseudo-nverse based estmaton method wth a quantzaton nose. Wthout nose the tradtonal methods predct better than the neural network because of the hghly lnear relatonshp between spectral sets used for tranng and predcton. o provde color constancy a Bayesan approach of the estmaton method s proposed by Branard and Freeman. 19 Snce the Bayesan approach s computatonally demandng, the submanfold method for spectral reflectance estmaton that s an ntermedate soluton between the Bayesan approach and lnear estmaton methods s descrbed by DCarlo and Wandell. 2 he method extends the lnear methods and ntroduces the addtonal term ncorporatng the nonlnearty of the data. he method uses a pece-wse lnear way to represent the nonlnear data structure and reduces the error value 12% n comparson wth a lnear method. It s mportant that the method partcularly reduces large lnear errors. he lmtaton of the method s that t needs a large tranng set and s nsuffcent when the data structure s a one-to-many mappng. he propertes of the methods consdered n ths paper are qute close to the submanfold approach 2 and one of the learnng algorthms based on Wener estmaton also gves a pece-wse lnear soluton. Recently, many advanced machne learnng technques usng neural networks and support vector machnes have been ntroduced and combned n the lbrares that are convenent for the purpose. For example, buldng the estmaton methods usng the ready-made machne learnng algorthms one can get theoretcally founded algorthms, 3

4 a unfed workflow for a current and future study, and a rch set of methods that provde flexblty for applcaton-orented research. In ths paper, the neural networks algorthms from the Netlab lbrary 21, 22 wll be used. hey nclude regresson, clusterng and pattern recognton methods. Many of these methods are densty models based on a lkelhood that s mportant for recognton and convenent for comparson wth other methods. In ths study, we statstcally analyze the reflectance spectra of color-patch sets of ol and watercolor pantngs wthout nose characterstcs, develop three machnelearnng based methods and compare them wth three tradtonal methods wth a synthetc data set and the real color-patch sets, as well. he tradtonal methods are lnear estmators based on low-dmensonal prncpal component analyss (PCA) approxmaton and Wener estmaton, and a nonlnear estmator based on multple regresson approxmaton. he machne learnng methods extend the tradtonal methods for estmatng a nonlnear data structure. hey nclude two nonlnear methods based on nonlnear prncpal component analyss and regresson analyss and the method usng pece-wse lnear Wener estmaton. he method utlzng nonlnear PCA and the method explotng pece-wse lnear Wener estmaton are novel methods. o develop an magng system, two measures are used for estmaton accuracy: spectral color dfference (RMSE) and colormetrc color dfference (CIE E 94 ). he former s better for archvng the spectral reflectance and the latter s better for evaluatng the appearance of the art pantngs under a specfc llumnaton to human observers. 4

5 he paper s arranged as follows: In the followng secton, we formulate the generalzed reconstructon of spectral reflectance from a multchannel mage n magng systems wth a reduced number of channels. Next, we descrbe three tradtonal methods and three machne learnng methods. hen we present the results of the statstcal analyss of the reflectance spectra of the color patches. Later on, an experment wth synthetc data and the reflectance spectra of the color patches s descrbed. Fnally, the expermental results are dscussed and concludng remarks are presented. Formulaton of the Spectral Reflecton Estmaton Fg. 1 shows the mage acquston system. he system conssts of the sngle chp hgh qualty CCD camera and the rotatng color wheel comprsng several color flters. he response v at poston ( x, y) of the CCD camera wth the th color flter s expressed as follows 3 : v ( x, y) = t ( λ ) E( λ) S( λ) r( x, y, λ) dλ + n ( x, y), = 1, K m, (1), where t (λ), E (λ), S (λ) and r ( x, y, λ) are the spectral transmttance of the th flter, the spectral radance of the llumnant, the spectral senstvty of the camera, and the spectral reflectance of a pantng, respectvely. n ( x, y) denotes addtve nose n the th channel mage and m denotes the total number of channels. 5

6 Fg. 1. he mage acquston system. For mathematcal convenence, each spectral characterstc wth l wavelengths s expressed as a vector or a matrx. Usng vector-matrx notaton, we can express Eq. (1) as follows: v ( x, y) = ES r( x, y) + n( x, y), (2) where denotes a transposton, v s an m 1 column vector representng the camera response, r s an l 1 column vector representng the spectral reflectance of the pantng, = t1, t 2, K, t m s an l m matrx n whch each column t represents the transmttance of the th flter, and E, S are the l l matrces that correspond to the spectral radance of the llumnant and the spectral senstvty of the CCD camera, respectvely. Further for the sake of smplcty, ( x, y) from v, r and n are omtted. Eq. (2) s rewrtten as an overall, lnear system matrx F = ES wth m l elements: v = Fr + n. (3) 6

7 he response of the spectral CCD camera v wthout a nose term s as follows: v = Fr. (4) We wll call the space spanned by r a spectral space and the space spanned by v a sensor space or subspace. he estmaton of reflectance spectra s obtaned as follows: r ˆ = Gv, (5) where G s a matrx dependng on the estmaton method used. In the next sectons, sx estmaton methods are consdered. radtonal Estmaton echnques hree approaches are usually used for spectral sensor desgn. he estmaton technques of reflectance spectra nclude: the method based on PCA (low-dmensonal approxmaton) (PCE), the method based on Wener estmaton (WE) and the method usng multple regresson approxmaton (MRE). 4 he Method Based on PCA Usng spectral reflectance of the tranng set r a covarance matrx s computed as follows: where E () s an expectaton operator. C = E(( r E( r))( r E( r)) ), (6) An egendecompston of the covarance matrx C determnes the matrx B = b, b, K, b 1 2 k, the columns of whch are k egenvectors correspondng to the frst k largest egenvalues. he spectral reflectance s approxmated as follows: r Bw, (7) where w s a vector of PCs, w = w 1, w2, K, w k and m k. 7

8 he spectral camera response gven by Eq. 4 can be presented by another expresson as follows 17 : v = FBw. (8) he prncpal components (PCs) are determned as follows: 1 w = ( FB) v. (9) Usng Eq. 7 and Eq. 9 the estmaton matrx G s as follows: 1 G = B( FB). (1) he estmate of the spectral reflectance of the pantng s as follows: 1 rˆ = Gv = B( FB) v, (11) where the data s centered by v v E(Fr) and means that the expresson on the rght s calculated and replaces the expresson on the left. Fnally, the mean value s added as follows: rˆ = rˆ + E( r). (12) Better accuracy of estmaton can be obtaned wth Wener estmaton, whch s consdered next. he Method Usng Wener Estmaton he Wener estmaton method mnmzes the overall average of the square error between the orgnal and estmated spectral reflectance. 3 For ths method, the correlaton matrces R rr of pantng spectra and nose consequently, the estmaton matrx s the followng 3 : R nn are frst computed, and 1 G = RrrF ( FR rrf + Rnn). (13) he estmate s as follows: rˆ 1 Gv = Rrr F ( FR rrf + R ) v. (14) = nn 8

9 If nose s not consdered, the estmaton matrx s as follows 3 : 1 G = R rrf ( FR rrf ). (15) And the estmate s as follows: 1 rˆ = Gv = R rr F ( FR rrf ) v. (16) In ths study, the Wener estmaton wthout consderaton to nose s used. he Wener estmaton gves good accuracy for lnear data. If the data s nonlnear, the technque based on multple regresson analyss s used. he Method Usng Multple Regresson Analyss In the case of nonlnear data, multple regresson analyss gves better results than Wener estmaton. 4 In the MRE method, the extended data matrx V of pantng spectra s frst defned through the data components and ther extended set of hgher-order terms as follows 4 : V = v, K, vm, v v, v v K, hgher order terms,, (17) K where denotes element-wse multplcaton. hen the estmaton matrx s gven as follows: 1 G = RV ( VV ), (18) where R s a matrx, the columns of whch are presented by n spectral samples gven by R = r, r, 2,, (19) 1 K r n Accordng to the lterature 4, the estmaton matrx G used n MRE s equal to the noseless varant of the Wener estmaton matrx. Fnally, 9

10 rˆ = GV = RV ( VV ) 1 V. (2) Owng to the fact that new advanced machne learnng algorthms are especally relevant for workng wth a nonlnear structure of data, the machne learnng technques are next dscussed for spectral estmaton. Machne Learnng Estmaton echnques Drawng analogy wth the tradtonal estmaton methods, three machne learnng technques are proposed. hey nclude the method based on regressve (nonlnear) PCA (RPCE), the method based on pece-wse lnear Wener estmaton (PLWE) and the method usng regresson analyss (RE). Eq.1 Eq.5 are vald for all machne learnng methods. he Method Based on Regressve PCA he spectral camera response s computed n the followng way: v = FBf w, θ ), (21) ( f where f () s a nonlnear vector-valued mappng functon and θ f s a parametrc vector. hen, PCs are defned by the followng equaton 1 w = h(( FB) v, ), (22) θ h where h() s an nverse functon, h () = f () 1, θ h s a parametrc vector and v v E(Fr). he mappng functon h () and parametrc vector θ h are computed usng a machne learnng algorthm for regresson. 21 In consequence, the spectral estmate of the pantng s as follows: 1

11 ˆ 1 r = Bh(( FB) v, ). (23) θ h Fnally, the mean value s added as follows: rˆ rˆ + E( r). (24) In practce, ths method nvolves a low-dmensonal subspace and a hgherdmensonal subspace ncludng the low-dmensonal subspace. For the lowdmensonal subspace, where w ( k ) = w, w, K, w, the mappng s as follows: 1 2 k w ( k ) 1 = h(( FB) v, θ h ) = ( FB) 1 v, (25) where v Fr E(Fr). For the hgher-dmensonal subspace, where ( p) ( k ) ( k + 1: p) w = w, w = w, w,, w, w, K, w, (26) 1 2 K k k + 1 the mappng s done for the hgher-order (or weak) PCs as follows: p w ( k + 1: p) = h(( FB) 1 v, θ h ) = h( w ( k ), θ). (27) hus the method uses the low-order real PCs and the hgher-order approxmated PCs. he Method Usng Pece-Wse Lnear Wener Estmaton In ths secton, the other machne learnng algorthm for pece-wse lnear Wener estmaton s dscussed. he man dea of the method s to separate the data structure nto parts whch are sutable for lnear approxmaton and each part s then estmated by usng the lnear Wener estmaton method. For data separaton, the clusterng algorthm s frst requred. he data s dvded nto several clusters v usng the Gaussan mxture model (GMM) 21 n a sensor space where s an ndex of the cluster. hen for the data of each cluster Wener estmaton s utlzed. Usng the labels of the data t s easy to compute the cluster covarance 11

12 matrx n the spectral doman needed for estmaton. When the th matrx follows: cluster covarance C of pantng spectra s known, the spectral estmate for the th cluster s as rˆ 1 = G v = CF ( FCF ) v, (28) where v v E Fr ). ( Fnally, the mean value s added as follows: r ˆ rˆ + E( r ). (29) he estmaton procedure s sequentally repeated for all clusters. he Method Usng Regresson Analyss he estmaton method based on the regresson analyss s smlar to the multple regresson approach. he dfference s that nonlnear mappng s used nstead of lnear mappng and the hgher-order terms are not syntheszed. For regresson analyss based on machne learnng the estmate s gven as follows: r ˆ = g( v, θ), (3) where g s a nonlnear vector-valued mappng functon and θ s a vector of parameters. hen, an th entry s defned as follows: r ˆ = ( v, θ). (31) g here are several regresson algorthms 21 but only the regresson method based on the radal bass functon (RBF) s used n ths study for all methods. he reason s that the RBF method s relatvely fast and performs well. 12

13 Addtonal echnques All machne learnng algorthms may need the addtonal technques that help n parameter adjustment. he regressve PCA method used n ths study s a technque whch combnes the PCA and nonlnear regresson methods. 23 In general, the ways utlzed n both approaches to detect the underlyng dmensonalty of the data can be combned. For PCA, ths s an analyss of the resdual energy dependng on a number of PCs. Furthermore, for regresson methods ths s Automatc Relevance Determnaton (ARD). 21 he ARD method defnes the statstcal dependence between the PCs, and n the case of the dependency between the tested components and a target component the tested components are relevant to approxmate the target component. However, ths technque wll not be used n ths study. For the regressve PCA the number of real PCs wll be gven and a number of approxmated PCs wll be used as a free parameter. he pece-wse lnear Wener estmaton approach needs to determne the number of lnear components for usng a clusterng procedure. hs s done based on the model selecton of the mxed dstrbuton. 24 After that the Gaussan mxture model 21 wth a gven number of clusters s used to extract lnear components. Statstcal Propertes of Reflectance Spectra For statstcal analyss of the spectral reflectance of pantngs we use fve sets of color patches of ol or watercolor pant as follows: set A, 336 patches of pant (reflectance of pant); set B, 6 patches of pant (urner acryl gouache); set C, 6 patches of pant 13

14 (urner golden acrylcs); set D, 91 patches of pant (Kusakabe ol pant) and set E, 18 patches of pant (Kusakabe haban). All sets were extracted from the standard object color spectral database constructed by the Spectral Characterstc Database Constructon Workng Group. 25 hese sets have a spectral range of 4-7 nm and samples are evenly taken at 1 nm. he set A s used for tranng the algorthms and the sets B-E are used for predcton of the spectral reflectance. herefore, lnear and nonlnear prncpal component analyss was carred out only for the set A. Accordng to a prevous publcaton 3, fve PCs of lnear PCA are good enough for accurate spectral estmaton. Hence the spectral set A and ts frst fve PCs that have a resdual energy of.16% are analyzed and shown n Fg. 2 and Fg. 3, respectvely. 1 Set A.8 Reflectance Wavelength, nm Fg. 2. Reflectance spectra of the set A of pant patches. 14

15 .5 Frst PC.5 Second PC hrd PC Fourth PC Ffth PC Wavelength, nm Wavelength, nm Fg. 3. Frst fve prncpal components of the set A of pant patches. If regressve PCA s appled to utlze the fve real PCs and several approxmated PCs of the set A, the average RMSE value of the spectral approxmaton s reduced (Fg. 4). hs llustrates the fact that there s a way to mprove the degree of accuracy for representng spectra by ncorporatng the nonlnearty of the data. Experment Synthetc Data In ths secton, the nonlnear dataset s frst syntheszed and then all methods for spectral estmaton are tested wth a synthetc set. It s assumed that one channel response s used whle the data smulatng spectra s two-dmensonal. he purpose of the test s to show the feasblty of the method to work wth data whch has a nonlnear structure. 15

16 9.5 x RMSE Number of components Fg. 4. he average RMSE of spectral approxmaton for the set A usng regressve PCA. he frst fve components are gven by PCA and the components 6-1 are approxmated by regressve PCA. hus two data components are generated for the test. he frst component x1 s 4 unformly dstrbuted n the range and another one s x ( x.5. 2 = 1 ) Fnally, a zero-mean Gaussan nose wth the standard devaton.7 was added to the generated components. he estmaton result of the synthetc data s presented n Fg. 5. A vector F, a vector b 1, that s a frst PCA egenvector from B and the curve correspondng to an underlyng subspace are shown n Fg. 5. he orgnal (syntheszed) data and the estmates for each method are shown by gray dots n Fg. 5. Although the WE method s superor to the PCE based method, the PCE and WE methods gve poor estmates for the data. he MRE, RPCE and PLWE methods are relatvely good for estmaton. he RE method gves the best result from among these methods. 16

17 Orgnal data PCE WE MRE x F 1 F 1 F 1 F b b b b x 1 RPCE PLWE RE x F 1 F 1 F b b b x x x 1 Fg. 5. he estmaton results for the synthetc data and dfferent estmaton methods. Real Data An experment was conducted wth sets A-E descrbed above. he set A s used for tranng whle the other sets are used for predcton. he spectral transmttance characterstcs of the separaton flters used n a CCD camera are gven n Fg. 6. he spectral senstvty of a CCD area sensor (Phase One 372 (horzontal pxels) 26 (vertcal pxels), 14 bts) s presented n Fg. 7. he llumnaton source s D65. 17

18 ransmttance BPB 42 SP 9 BPB 55 BPB 5 BPB Wavelength, nm Fg. 6. he spectral transmttance characterstcs of the flters. 1.8 Senstvty Wavelength, nm Fg. 7. he spectral senstvty of the camera. he parameters used n the test are the followng: he fve PCs are exploted for PCE and RPCE. In addton, the RPCE approach uses the PCs approxmatng the real sxth, seventh, eghth and nnth PCs. For the PLWE method a mxture of Gaussan components s used for clusterng where the number of components s defned n a test based on the model selecton of the mxed dstrbuton. he MRE technque uses the terms begnnng wth the frst-order to the second-order ones. For the RE method, regresson s based on the radal bass functon usng the Gaussan functon. 2 neurons and 7 teratons are used n ths case. 18

19 A varatonal Bayesan model selecton method for the mxture dstrbuton 24 n the sensor space defnes the number of components for the PLWE method. For ths, the program s rerun ten tmes. he results are presented n able 1 where the frst row shows the test number and the second row shows the number of components determned by the algorthm. Fg. 8 llustrates the varatonal lkelhood bound over the model selecton of 336 pantng spectra (set A). Intally, the model has ten Gaussans. he vertcal lnes show the removal of the components from the model. Fnally, two components are selected. able 1. he number of components for pece-wse lnear Wener estmaton est number Number of components Lakelhood bound Mxture components Iteratons Fg. 8. he varatonal lkelhood bound over the model selecton of 336 pantng spectra (set A). 19

20 If the estmaton values of spectral reflectance are less than zero or greater than one then they are equalzed to zero or one, respectvely In able 2 and able 3, the average and maxmum RMSE values for each set are gven for the tradtonal methods and methods based on machne learnng algorthms. able 2. he average and maxmum (n parentheses) RMSE values for PCE, WE and MRE PCE WE MRE Set A.516 (.2458).155 (.1633).123 (.1159) Set B.836 (.3952).346 (.1712).324 (.1732) Set C.889 (.3469).466 (.2478).397 (.2158) Set D.917 (.483).43 (.234).352 (.275) Set E.917 (.3136).33 (.1416).281 (.1199) able 3. he average and maxmum (n parentheses) RMSE values for RPCE, PLWE and RE RPCE PLWE RE Set A.512 (.2447).142 (.1522).123 (.147) Set B.834 (.3928).343 (.1683).315 (.1731) Set C.887 (.3452).45 (.235).379 (.21) Set D.912 (.466).376 (.229).349 (.1992) Set E.91 (.3122).339 (.1185).275 (.162) 2

21 In able 4 and able 5, the average and maxmum CIE E 94 values for each set are gven for the tradtonal methods and methods based on machne learnng algorthms. able 4. he average and maxmum (n parentheses) CIE E 94 values for PCE, WE and MRE PCE WE MRE Set A.72 (13.65).17 (4.3).15 (1.68) Set B 2.96 (21.).58 (2.84).54 (2.13) Set C 2.36 (15.42).8 (4.8).59 (4.21) Set D 2.43 (19.24).71 (5.18).55 (3.37) Set E 1.32 (3.57).37 (2.34).31 (1.18) able 5. he average and maxmum (n parentheses) CIE E 94 values for RPCE, PLWE and RE RPCE PLWE RE Set A.81 (14.89).16 (3.46).17 (3.16) Set B 3.34 (23.15).67 (2.65).59 (2.65) Set C 2.51 (14.9) 1.33 (8.47).82 (3.47) Set D 2.71 (2.86).8623 (8.19).74 (2.92) Set E 1.89 (5.14).57 (2.).71 (2.79) In general, the results presented n able 2 - able 5 demonstrate that for the RMSE values the machne learnng methods gve slghtly better results than ther tradtonal opposte methods whle the tradtonal methods have smaller CIE E 94 values. he excepton s the RE method whch has better predcton n comparson wth the other methods for the maxmal error of the color dfference. 21

22 he methods are also tested usng computatonal tme. he CPU tme n seconds for the set A s presented n able 6. For the algorthms, the CPU tme s gven separately for tranng (upper row) and predcton (lower row). In able 6, zero values are gven for the CPU tme, whch s very small (ths corresponds to several matrx-vector multplcatons). Matlab 6.5, the Intel Pentum III Processor, 166 MHz and 248 MB of RAM are used n the test. he test shows that the tradtonal methods are faster than the machne learnng methods. However, the predcton tme for the machne learnng methods s relatvely short. able 6. he CPU tme n seconds PCE WE MRE RPCE PLWE RE o see whether any nonlnearty s presented n the estmated spectra we measure the average RMSE value after estmaton of spectral reflectance usng PCA and RPCA. he results are shown n able 7 for PCA wth the fve PCs (upper number) and for RPCA wth the fve real PCs and fve approxmated (from sx to ten) PCs (lower number). hen, the rato between these two RMSE values s determned and presented n able 8. From able 8, one can see that the RE and RPCE methods have rato values close to the orgnal data set. he MRE and PLWE methods gve results whch are farther from the orgnal data set. he PCE and WE rato values are the most dfferent from the orgnal data n comparson wth the other methods. 22

23 able 7. he average RMSE value after spectral estmaton for PCA wth the fve PCs (upper number) and for RPCA wth the fve real components and fve approxmated components (lower number). Set A PCE WE MRE RPCE PLWE RE able 8. he rato between the RMSE values for PCA and RPCA Set A PCE WE MRE RPCE PLWE RE From among the tradtonal methods the method based on MRE produces the best result. he method has small RMSE and CIE E 94 values n the tranng set and sets used for predcton. Whle the RMSE values for all machne learnng methods are slghtly better n comparson wth the tradtonal methods, the CIE E 94 values of the methods based on machne learnng except the RE method are hgher. he overall means of average color dfferences for the tradtonal methods are 1.95 (PCE),.52 (WE) and.42 (MRE) and for the learnng methods 2.25 (RPCE),.65 (PLWE) and.6 (RE). hus, the color dfferences usng the machne learnng methods are smaller than the dfferences between the tradtonal methods. he RE method ncorporates nonlnearty of data that s clearly seen from able 8. he generalzaton of the data gven by the RE method s very good n comparson wth the other methods. hs follows from predctng the maxmum CIE E 94 values. However, gven the processng and executon tmes the MRE method gves a better average and n two out of fve cases smaller maxmum color dfference errors than the RE method. Although 23

24 the tradtonal methods are less tme consumng than the machne learnng methods, the predcton tme for the learnng methods s short enough. In general, the tradtonal methods look more desrable than the learnng methods. hs s contrary to the ntal result shown n Fg. 5 where the learnng methods are superor to the tradtonal methods. hs can be explaned as follows. In ths study the sensor space (subspace) dmensonalty s defned by the fve gven flters. Although the subspace s not optmal (close to optmal) ts dmensonalty s rather hgh. Recently, t was shown that for reflectance spectra the dmensonalty of the nonlnear subspace s approxmately three. 26 hus, one can expect that for spectral magng systems havng the low dmensonal sensor space or fewer channels the learnng based methods are more effcent. We wll consder ths problem n a future study. Conclusons We have compared the methods for estmatng the spectral reflectance of art pantngs for the development of spectral color magng systems. hree tradtonal methods and three methods based on machne learnng for spectral reflectance estmaton of pant were utlzed. he tradtonal methods nclude two lnear methods the method based on PCA and the method based on Wener estmaton and one method usng multple regresson analyss. We ntroduced two novel machne learnng methods utlzng regressve PCA and pece-wse lnear Wener estmaton. hus, the machne learnng methods nclude two methods workng wth a global nonlnear data structure the method based on regressve PCA and the method based on regresson analyss and the method usng pece-wse lnear Wener estmaton. Smlarly to the submanfold method 2, the learnng methods used are between the lnear and Bayesan approaches 24

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